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Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 163-212
DOI: https://doi.org/10.7151/dmgaa.1071

ADJOINTNESS BETWEEN THEORIES AND STRICT THEORIES

Hans-Jürgen Vogel

Institute of Mathematics, University of Potsdam
PF 60 15 53, D-14415 Potsdam, Germany
e-mail: vogel@rz.uni-potsdam.de
         hans-juergen.vogel@freenet.de

Dedicated to Prof. Dr. habil. Klaus Denecke on the occasion of his 60th birthday

Abstract

The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols.

If the object class of a dhts-category forms even a monoid with unit element I and zero element O, then one has a strict partial theory.

In this paper is shown that every J-sorted partial theory corresponds in a natural manner to a J-sorted strict partial theory via a strongly d-monoidal functor. Moreover, there is a pair of adjoint functors between the category of all J-sorted theories and the category of all corresponding J-sorted strict theories.

This investigation needs an axiomatic characterization of the fundamental properties of the category Par of all partial function between arbitrary sets and this characterization leads to the concept of dhts- and dhth∇ s-categories, respectively (see [5], [11], [13]).

Keywords: symmetric monoidal category, dhts-category, partial theory, adjoint functor.

2000 Mathematics Subject Classification: 18D10, 18D20, 18D99, 18A25, 08A55, 08C05, 08A02.

References

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[14]	H.-J. Vogel, On functors between dht∇ -symmetric categories, Discuss. Math.-Algebra & Stochastic Methods, 18 (1998), 131-147.
[15]	H.-J. Vogel, On properties of dht∇ -symmetric categories, Contributions to General Algebra 11 (1999), 211-223.

Received 25 July 2003
Revised 21 October, 2003